Ratio and Proportion – Basic 6 Mathematics Lesson Note (Second Term)

What is Ratio?

A ratio compares two or more quantities. It shows how many times one quantity is contained in another.

Written as: 3:4 or 3/4 Read as: “3 to 4”

Example: Boys to Girls

In a class of 30 students, 18 are boys and 12 are girls. Ratio of boys to girls = 18:12 = 3:2

Simplifying Ratios

Method: Divide both numbers by their H.C.F

Example: 24:36 H.C.F of 24 and 36 = 12 24 ÷ 12 = 2, 36 ÷ 12 = 3 Simplified ratio = 2:3

Exercise A – Simplifying Ratios

Simplify these ratios:

1. 15:20 = ______
2. 24:32 = ______
3. 45:30 = ______
4. 28:42 = ______
5. 36:48:60 = ______

Sharing in Given Ratios

Example: Share ₦1,500 between Kemi and Tunde in ratio 2:3

Step 1: Total ratio = 2 + 3 = 5 parts Step 2: Kemi gets 2/5 × ₦1,500 = ₦600 Step 3: Tunde gets 3/5 × ₦1,500 = ₦900

Exercise B – Sharing in Ratios

Solve these sharing problems:

1. Share ₦2,400 between two friends in ratio 3:5 First friend: ₦______ Second friend: ₦______
2. Divide 240 books among three classes in ratio 2:3:4 Class A: ______ Class B: ______ Class C: ______
3. Share 150 oranges between Primary 5 and Primary 6 in ratio 2:3 Primary 5: ______ Primary 6: ______

Direct Proportion

What is Direct Proportion?

When two quantities increase or decrease together at the same rate, they are in direct proportion.

Example: More workers, more work done Formula: If a ∝ b, then a/b = k (constant)

Direct Proportion Method

Example: 5 books cost ₦250. How much do 8 books cost?

Method 1: Unitary Method 5 books cost ₦250 1 book costs ₦250 ÷ 5 = ₦50 8 books cost ₦50 × 8 = ₦400

Method 2: Proportion 5 books : ₦250 = 8 books : x 5/250 = 8/x x = (8 × 250) ÷ 5 = ₦400

Exercise C – Direct Proportion

Solve these problems:

1. If 6 pencils cost ₦180, how much do 10 pencils cost? Answer: ₦______
2. A car travels 120 km in 2 hours. How far will it travel in 5 hours at the same speed? Answer: ______ km
3. 12 workers can complete a task in 8 days. How long will 16 workers take? Answer: ______ days
4. If 4 kg of rice costs ₦1,600, how much does 7 kg cost? Answer: ₦______
5. A tap fills 15 buckets in 30 minutes. How many buckets will it fill in 50 minutes? Answer: ______ buckets

Inverse Proportion

What is Inverse Proportion?

When one quantity increases and the other decreases at the same rate, they are in inverse proportion.

Example: More workers, less time needed Formula: If a ∝ 1/b, then a × b = k (constant)

Inverse Proportion Method

Example: 8 workers can build a wall in 12 days. How many days will 6 workers take?

Method: Workers × Days = Constant 8 × 12 = 96 6 × Days = 96 Days = 96 ÷ 6 = 16 days

Exercise D – Inverse Proportion

Solve these problems:

1. 24 workers can finish a project in 15 days. How long will 18 workers take? Answer: ______ days
2. A car traveling at 60 km/h takes 4 hours for a journey. How long will it take at 80 km/h? Answer: ______ hours
3. 12 cows eat grass in a field for 20 days. How long will the grass last for 15 cows? Answer: ______ days
4. If 20 pumps can empty a pool in 6 hours, how many pumps are needed to empty it in 4 hours? Answer: ______ pumps
5. A farmer has food for 50 chickens for 24 days. How long will the food last for 40 chickens? Answer: ______ days

Real Life Problems on Ratio and Proportion

Mixed Problems

Exercise E – School Problems

Solve these school-related problems:

1. Recipe Scaling: A recipe for 4 people needs 2 cups of flour. How much flour is needed for 10 people? Answer: ______ cups
2. Class Ratio: In Primary 6A, the ratio of boys to girls is 3:2. If there are 15 boys, how many girls are there? Answer: ______ girls
3. Sports Team: A football team wins 4 out of every 5 matches. If they play 25 matches, how many will they win? Answer: ______ matches

Exercise F – Money and Business

Solve these money problems:

1. Partnership Business: Three friends start a business. Ade contributes ₦200,000, Bola ₦300,000, and Chidi ₦100,000. If they make ₦180,000 profit, how should it be shared? Ade: ₦______ Bola: ₦______ Chidi: ₦______
2. Exchange Rate: If $1 = ₦750, how many naira will you get for $15? Answer: ₦______
3. Discount Shopping: A shop gives 3 free items for every 12 items bought. If you buy 48 items, how many free items do you get? Answer: ______ free items

Exercise G – Time and Work

Solve these work problems:

1. Construction Project: 15 men can build a house in 40 days. How many men are needed to build it in 25 days? Answer: ______ men
2. Farm Work: 8 tractors can plough a field in 6 hours. How long will 12 tractors take? Answer: ______ hours
3. Water Supply: A water tank lasts 30 families for 12 days. How long will it last for 20 families? Answer: ______ days

Quantitative Reasoning

Pattern Recognition in Ratios

Exercise H – Ratio Patterns

Complete these ratio patterns:

1. 3:6, 6:12, 9:18, _____, _____
2. 1:4, 2:8, 3:12, _____, _____
3. 2:1, 4:2, 6:3, _____, _____

Exercise I – Proportion Reasoning

Use logical thinking to solve:

1. Speed and Time: If speed doubles, time becomes _______ (half/double)
2. Workers and Days: If the number of workers increases, days needed _______ (increase/decrease)
3. Price and Quantity: If you buy more items, total cost _______ (increases/decreases)

Exercise J – Problem Analysis

Identify if these are direct or inverse proportion:

1. More books, more weight = _______ proportion
2. More speed, less time = _______ proportion
3. More workers, more salary paid = _______ proportion
4. More people, less food per person = _______ proportion

Answer Key

Exercise A:

1. 3:4
2. 3:4
3. 3:2
4. 2:3
5. 3:4:5

Exercise B:

1. ₦900, ₦1,500
2. 53, 80, 107
3. 60, 90

Exercise C:

1. ₦300
2. 300 km
3. 6 days
4. ₦2,800
5. 25 buckets

Exercise D:

1. 20 days
2. 3 hours
3. 16 days
4. 30 pumps
5. 30 days

Exercise E:

1. 5 cups
2. 10 girls
3. 20 matches

Exercise F:

1. Ade: ₦60,000, Bola: ₦90,000, Chidi: ₦30,000
2. ₦11,250
3. 12 free items

Exercise G:

1. 24 men
2. 4 hours
3. 18 days

Exercise H:

1. 12:24, 15:30
2. 4:16, 5:20
3. 8:4, 10:5

Exercise I:

1. half
2. decrease
3. increases

Exercise J:

1. direct
2. inverse
3. direct
4. inverse